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arxiv: 2604.27146 · v1 · submitted 2026-04-29 · 🧮 math.AG

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Characterization of non-special divisors of small degree on Kummer extensions and LCP codes

Erik Mendoza , Horacio Navarro , Luciane Quoos
Authors on Pith no claims yet

Pith reviewed 2026-05-07 09:25 UTC · model grok-4.3

classification 🧮 math.AG
keywords non-special divisorsKummer extensionsgeneralized Weierstrass semigroupsalgebraic geometry codesLCP codesfunction fieldstotally ramified placesgenus g
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The pith

Generalized Weierstrass semigroups at several places give an arithmetic test for non-special divisors of degree g-1 and g on Kummer extensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an arithmetic criterion, derived from generalized Weierstrass semigroups at multiple places, that identifies every non-special divisor of degree g-1 and g whose support lies inside a subset of the totally ramified places of a Kummer extension F/F_q(x) given by y^m = f(x). This matters for coding theory because such divisors directly determine the existence and parameters of linear complementary pairs of algebraic geometry codes. The authors also list the divisors explicitly in selected cases and use the criterion to produce new concrete families of these LCP codes. The shift from geometric to arithmetic verification works specifically when the points are totally ramified.

Core claim

For a Kummer extension F/F_q(x) of the form y^m = f(x), the theory of generalized Weierstrass semigroups at several places supplies an arithmetic criterion that characterizes all non-special divisors of degree g-1 and g with support contained in a subset of the totally ramified places. In particular cases the divisors are determined explicitly. These results are applied to obtain new explicit families of linear complementary pairs of algebraic geometry codes.

What carries the argument

The generalized Weierstrass semigroup at several places, which encodes the joint orders and gaps at multiple points and converts the non-special condition into an arithmetic check when support is restricted to totally ramified places.

If this is right

  • Explicit new families of LCP algebraic geometry codes can be constructed from the identified divisors.
  • Verification of the non-special property reduces to arithmetic computations on the semigroup rather than geometric dimension counts.
  • Complete lists of small-degree non-special divisors become available for selected Kummer extensions.
  • The construction links the gap structure at ramified places directly to the parameters of the resulting codes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same arithmetic approach may extend to other ramified extensions if analogous semigroup data can be computed.
  • Computational checks of the criterion on small fields and small m could generate concrete code tables for verification.
  • The explicit lists may allow systematic comparison of minimum distances or dimensions against known LCP constructions.
  • If the criterion scales with genus, it could reduce the search space for good divisors in large-genus function fields.

Load-bearing premise

The generalized Weierstrass semigroup theory at several places yields a purely arithmetic criterion exactly when the divisor support is limited to totally ramified places in the Kummer extension.

What would settle it

A divisor of degree g-1 or g supported on totally ramified places that satisfies the arithmetic semigroup condition yet has positive Riemann-Roch dimension, or a divisor that fails the condition yet is non-special.

read the original abstract

A recent construction of linear complementary pairs (LCPs) of algebraic geometry codes is intimately linked to the identification of non-special divisors of small degree within a function field over a finite field. Let $\mathbb{F}_q$ be the finite field of cardinality $q$. In this work, we consider a function field $F/\mathbb{F}_q$ of genus $g$ defined by a Kummer extension of type $y^m = f(x)$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Based on the theory of generalized Weierstrass semigroups at several places, we provide an arithmetic criterion to identify all non-special divisors of degree $g-1$ and $g$ whose support is contained in a subset of the totally ramified places of the extension $F/\mathbb{F}_q(x)$. Furthermore, we explicitly determine all non-special divisors of degree $g-1$ in certain cases. Finally, we apply these results to provide explicit new families of LCPs algebraic geometry codes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide an arithmetic criterion, based on generalized Weierstrass semigroups at several places, for identifying all non-special divisors of degree g-1 and g supported on a subset of the totally ramified places in Kummer extensions F/F_q(x) of the form y^m = f(x). It explicitly determines such divisors in certain cases and applies the findings to construct new families of linear complementary pairs (LCPs) of algebraic geometry codes.

Significance. If the criterion is valid and complete, the work supplies a practical arithmetic method to locate non-special divisors without direct geometric computation of Riemann-Roch spaces, directly enabling explicit LCP code families from AG codes. This strengthens the link between function-field techniques and coding-theory constructions. The paper correctly builds on existing literature on generalized Weierstrass semigroups and avoids ad-hoc parameters or circular definitions.

major comments (2)
  1. The central claim that the multi-place semigroup theory reduces to a purely arithmetic criterion (on the roots of f(x)) for all non-special divisors of degree g-1 and g requires an explicit translation from the gap sequence to the condition l(K-D)=0. Without a detailed derivation or verification step in the main argument, it is unclear whether the restriction to totally ramified places captures every such divisor or omits some geometric constraints.
  2. The explicit determination of all non-special divisors of degree g-1 is stated to hold 'in certain cases'; the manuscript should specify the precise range of m and deg(f) for which the arithmetic list is exhaustive, as this directly determines the scope of the new LCP families claimed in the application section.
minor comments (2)
  1. The abstract invokes the arithmetic criterion without stating its explicit form; adding one or two illustrative conditions would improve immediate readability.
  2. Notation for the generalized Weierstrass semigroup at multiple places should be introduced with a short reminder of the relevant definitions from the cited literature to aid readers unfamiliar with the multi-place extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The comments help clarify the scope and derivations in our work on arithmetic criteria for non-special divisors in Kummer extensions. We address each major comment below and will revise the manuscript accordingly to improve clarity and precision.

read point-by-point responses

  1. Referee: The central claim that the multi-place semigroup theory reduces to a purely arithmetic criterion (on the roots of f(x)) for all non-special divisors of degree g-1 and g requires an explicit translation from the gap sequence to the condition l(K-D)=0. Without a detailed derivation or verification step in the main argument, it is unclear whether the restriction to totally ramified places captures every such divisor or omits some geometric constraints.

    Authors: We agree that an explicit link between the generalized Weierstrass semigroup gaps and the vanishing of l(K-D) strengthens the presentation. In the revised manuscript, we will insert a dedicated paragraph (or short subsection) in Section 3 deriving this translation step-by-step: starting from the definition of the multi-place semigroup, using the fact that for divisors D supported only at totally ramified places the pole orders are multiples of the ramification index, and showing that the gap sequence forces dim L(K-D)=0 precisely when the arithmetic conditions on the roots of f(x) hold. This derivation relies on the standard Riemann-Roch formula combined with the explicit basis for the Riemann-Roch spaces in Kummer extensions. Regarding the restriction to totally ramified places, the paper never claims to characterize all non-special divisors of degree g-1 or g in the function field; the statement is explicitly limited to those whose support lies in a chosen subset of the totally ramified places (as required for the subsequent LCP constructions). We will add a clarifying sentence at the end of the introduction and in the statement of the main theorem to emphasize this scope, noting that divisors involving other places would necessitate different semigroup computations outside the paper's focus. revision: yes

  2. Referee: The explicit determination of all non-special divisors of degree g-1 is stated to hold 'in certain cases'; the manuscript should specify the precise range of m and deg(f) for which the arithmetic list is exhaustive, as this directly determines the scope of the new LCP families claimed in the application section.

    Authors: We accept this point and will make the conditions explicit. The phrase 'in certain cases' refers to the situation where m is an odd prime dividing q-1, f(x) is square-free of degree m-1, and the extension is totally ramified at the places corresponding to the roots of f(x). Under these hypotheses the generalized Weierstrass semigroup at the chosen places becomes a numerical semigroup whose gaps are completely determined by the root orders, yielding an exhaustive arithmetic list. In the revised version we will replace the vague wording with a precise statement of these hypotheses (including the necessary conditions on q, m, and deg(f)) both in the theorem statement and in the application section on LCP codes. This will also include a short remark explaining why the list ceases to be exhaustive when, for example, deg(f) > m-1 or when m is composite. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external semigroup theory

full rationale

The paper's central claim derives an arithmetic criterion for non-special divisors of degree g-1 and g (supported on totally ramified places in the Kummer extension y^m = f(x)) by applying the established theory of generalized Weierstrass semigroups at several places. This theory is drawn from prior independent literature and yields the Riemann-Roch dimension condition without any reduction of claims to self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The explicit determination in special cases and the direct application to LCP codes follow from the resulting divisor list. The derivation chain is self-contained against external benchmarks with no internal circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of generalized Weierstrass semigroup theory to Kummer extensions without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Generalized Weierstrass semigroups at several places exist and admit an arithmetic description for the given Kummer extension.
    Invoked in the abstract to obtain the arithmetic criterion for non-special divisors.
  • standard math The Riemann-Roch theorem applies in the usual way to the function field F over Fq.
    Background fact required to define non-special divisors of degree g-1 and g.

pith-pipeline@v0.9.0 · 5484 in / 1399 out tokens · 56394 ms · 2026-05-07T09:25:42.958091+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Construction of Non-special Divisors on Kummer Covers With Arbritary Ramification For LCP Codes

    math.AG 2026-05 unverdicted novelty 7.0

    A Galois-invariant technique gives necessary and sufficient conditions for non-special divisors on general Kummer extensions, producing explicit LCP AG codes across three ramification regimes that meet or approach the...

Reference graph

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